Updated 2:23 p.m. | Republicans said work will continue to reach a deal to raise the debt limit and end the government shutdown after meeting with President Barack Obama.

Sen. Orrin G. Hatch, R-Utah, said the conversation with the president was "wide-ranging," but he said it was clear both sides had demands the other side would be unwilling to meet.

Hatch, the top Republican on the Finance Committee, said the GOP is still in a good position against the Democrats' health care law, even though the Republican decision to shut down the government over defunding the law has made it more popular than it was before the GOP did so.

As far as what Obama is looking for, Hatch said he would want to roll back the across-the-board discretionary spending cuts known as the sequester the two parties agreed to at the time of the last big budget fight over the debt limit in 2011.

"In all honesty, a lot of us would like to be able to solve this problem for the good of the country, but you can't ... he'd like to get rid of sequestration, to be honest with you, that's something that isn't going to happen," Hatch said, before adding that he had "hope" the government could re-open next week.

Like many of his colleagues, Minority Leader Mitch McConnell, R-Ky., left the meeting focused on the way forward.

"Now we're back here to actually work on trying to get a solution on a bipartisan basis. As you know, there are a number of different discussions going on, and now that we're back up here and back at work, hopefully we can find a way forward," McConnell said before a meeting with his conference back at the Capitol.

Talks that have little to do with the House and White House continued in the Senate Friday, before and after the meeting with the president.

According to multiple sources in both parties, cross-party conversations have ticked up in recent days, with Sen. Charles E. Schumer, D-N.Y., fielding multiple phone calls from Republicans to listen to the ideas being floated across the aisle and give insight into what might ultimately be acceptable for Democrats. One of the more discussed — and perhaps politically feasible — ideas is repealing the medical device tax, a move that's already received a filibuster-proof majority of votes in the Senate (if there's an acceptable offset).

Arizona Republican John McCain, who has also been involved in the talks, told reporters at the Capitol after the meeting that he couldn't pinpoint when a deal might be reached.

"This things always end suddenly. All of a sudden there's critical mass. This, I think, was an important step in the right direction because we made ourselves very clear and the president made himself very clear," McCain said. "Can I say this is significant progress? No, I can't, but I can say it was a very useful exercise that we had this discussion. I wish we would've had it years ago."

Sen. Roy Blunt said he didn't take much away from the trip down Pennsylvania Avenue.

"The president reinterated many times that he doesn't want to negotiate on the debt ceiling. I think there's a failure there to understand that there is no normal process. You know, the president said a couple of different times you shouldn't use moments like this that you couldn't get in the normal process," the Missouri Republican said. "But the truth is there is no normal process, and so you have to use the moment you can."

McCain had outlined a scenario Thursday on CNBC before the House GOP leaders returned from meeting with Obama that could suggest one possible endgame — a scenario in which the House accepts the terms of a Senate agreement.

"The House, I don't think, will do the bidding of the president of the United States," McCain said. "But if there is a proposal that's agreed to here in the Senate by Republicans and Democrats, and the president probably would be hard pressed to disagree with that, if both members — the majority of members of both parties agreed to it."

Majority Leader Harry Reid, D-Nev., on Friday signaled resistance to a deal that would only forestall the debt limit for six weeks.

"Democrats believe a six-week delay of a catastrophic default is not enough to give the economy the confidence it needs to continue growing and recovering," said Reid, cautioning against setting another deadline at the start of the holiday shopping season.